= Solution
A zero-mean <autoregressive process of order one> satisfies
$$
Y_t=\alpha Y_{t-1}+\varepsilon_t,
$$
where $\varepsilon_t$ is <white noise> with <variance> $\sigma_\varepsilon^2>0$. A <causal time series> uses present and past innovations only. For $|\alpha|<1$ the series
$$
Y_t=\sum_{j=0}^\infty\alpha^j\varepsilon_{t-j}
$$
has <mean-square convergence> because $\sum_j\alpha^{2j}<\infty$, and it supplies the causal <weakly stationary process>. Its <variance> is $\sigma_\varepsilon^2/(1-\alpha^2)$. For $h\ge0$, only the common innovations contribute to the <autocovariance>:
$$
\gamma(h)=\sigma_\varepsilon^2\sum_{j=0}^\infty\alpha^{h+j}\alpha^j=\frac{\sigma_\varepsilon^2\alpha^h}{1-\alpha^2}.
$$
Evenness then gives
$$
\boxed{\gamma(h)=\frac{\sigma_\varepsilon^2\alpha^{|h|}}{1-\alpha^2},\qquad \rho(h)=\alpha^{|h|}.}
$$
When $\alpha<0$, the signs alternate; when $\alpha=0$, the process is white noise. With nondegenerate innovations, $|\alpha|\ge1$ does not give a causal finite-variance stationary solution. These <covariance> calculations need uncorrelated innovations, not Gaussianity; Gaussian innovations additionally yield a Gaussian process.
Back to article page
